2020/10/09 by Rukmini Dey, Dey, Rukmini, Kohinoor Ghosh +3 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Advanced Mathematical Theories and Applications #Advanced Topics in Algebra #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematics and Applications #Scientific Research and Discoveries #Soft Condensed Matter (cond-mat.soft)
paper · pdf · doi:10.48550/arxiv.2010.04405
openalex publication_date 2020/10/09 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
We show that the height function of Scherk's second surface decomposes into a\nfinite sum of scaled and translated versions of itself, using an Euler\nRamanujan identity. A similar result appears in R. Kamien's work on liquid\ncrystals where he shows (using an Euler-Ramanujan identity) that the Scherk's\nfirst surface decomposes into a finite sum of scaled and translated versions of\nitself. We give another finite decomposition of the height function of the\nScherk's first surface in terms of translated helicoids and scaled and\ntranslated Scherk's first surface. We give some more examples, for instance a\n(complex) maximal surface and a (complex) BI soliton. We then show, using the\nWeierstrass-Enneper representation of minimal (maximal) surfaces, that one can\ndecompose the height function of a minimal (maximal) surface into finite sums\nof height functions of surfaces which, upon change of coordinates, turn out to\nbe minimal (maximal) surfaces, each minimal (maximal) w.r.t. to its own new\ncoordinates. We then exhibit a general property of minimal surfaces, maximal\nsurfaces, timelike minimal surfaces and Born-Infeld soliton surfaces that their\nlocal height functions z=Z(x,y) split into finite sum of scaled and\ntranslated versions of functions of the same form. Upto scaling these new\nfunctions are height functions of the minimal surfaces, maximal surfaces,\ntimelike minimal surfaces and Born-Infeld soliton surfaces respectively.\nLastly, we exhibit a foliation of mathbb R3 minus certain lines by\nshifted helicoids (which appear in one of the Euler-Ramanujan identities).\n